Programming Exercise 1

This exercise will get you started with programming in C. For this exercise, you will need to download the template ex1.c, and enter your code in the methods (functions) provided. You are welcome to create your own methods if you like. You will also need to download two additional C programs - the file uniform.c and the file ex1test.c into the same directory. Do not make changes to uniform.c. You may change or add test code to ex1test.c if you like. Both files need to be in the same directory as your ex1.c. Before you write your first line of code, you ought to try compiling the template as follows:

  gcc -o ex1 ex1test.c ex1.c uniform.c -lm

This will create an executable called ex1 that you can execute (on Unix):

  ./ex1

In this exercise you will estimate two probabilities:

Given a number M, the probability that in any random group of M people, at least two people share a birthday.
Given numbers M and K, the probability that in any random group of M people, you find a subset of K people whose birthdays fall on K consecutive days.

This is why there are two methods called sharedBirthdayProbability (for the first problem) and consecutiveBirthdaysProbability (for the second problem) that you will implement. Let's look at the signature of each of these:

First one:
```
double sharedBirthdayProbability (int numPeople, int numTrials)
{
  /* INSERT YOUR CODE HERE */
}
  
```
This method takes in two parameters, the number of people (M) and the number of random trials that you will use in your estimate.
Second one:
```
double consecutiveBirthdaysProbability (int numPeople, int numConsecutive, int numTrials)
{
  /* INSERT YOUR CODE HERE */
}
  
```
This method takes in three parameters: the number of people (M), the number of consecutive days (K) and the number of trials to be used in the estimate.

What do we mean by "number of trials"? First, consider how we are going to solve each problem:

Let's examine one approach to solving the first problem, and let's consider one trial for now. Create an array of length 365 where each entry is initialized to zero. For each of M people, pick a random day (a random spot in the array) as a birthday. If any spot was picked by two or more people, we know that a birthday has been shared. Of course, if we perform just one trial, we could be lucky and observe a shared birthday or we could be unlucky and observe no shared birthdays. So, to get an accurate estimate, we need to repeat this experiment many times (perhaps 10000 times) and take the average. In other words, we need to compute the fraction of trials (out of 10000 trials) in which a shared birthday occurred.
The second problem can be solved in a similar way, except that you have to look for K consecutive birthdays. Note: in this problem you have to worry about "wrap around": for example (when K=3), the three days could be Dec 31, Jan 1 and Jan 2 (and should therefore count).

How does one pick a random day for a particular person? You will use the discrete_uniform() method in the file uniform.c. The call

    i = discrete_uniform (1, 10);

returns an integer selected at random between 1 and 10 inclusive. You can easily modify this to return a randomly selected day. For practice, you might want to write the program in Java as well. To use the equivalent program in Java, you will need to download the file UniformRandom.java and use the following method:

    i = (int) UniformRandom.uniform (1, 10);

Deliverables:

The source code for your program. Please write all your code in the same file (ex1.c).
The following estimates:
- Shared-birthday probability for M=10.
- What is the least value of M for which the shared-birthday probability is larger than 0.5?
- The consecutive-birthdays probability for M=10 and K=3.
- With K=3, what is the least value of M for which the consecutive-birthdays probability is larger than 0.5?
Put these numbers (manually) in a plain text file called results.txt in your directory. For each estimate, use at least 10,000 trials of the experiment.
You do not need to write/submit four programs. Simply change the parameter values for four executions.
NOTE: Your code will be graded on both correctness and documentation (comments).

Submission:

You have already created a directory called ex1 directory in your Unix account, following these instructions for setting up your Unix account and for submitting work.
Make a zip of this folder and upload into Blackboard in the right place.